Probabilistic Analysis of Message Forwarding

Louise Moser and Michael Melliar-SmithUniversity of California, Santa Barbara

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Message Forwarding

Direct multicasting of messages to many nodes is expensive and might be infeasible, if the network is large

Source node transmits its message to a small number of nodes• Each such node retransmits the message to other nodes, chosen at

random• Spreads the load across multiple nodes• Introduce a probability of message forwarding• Limit the number of levels of message forwarding

It is easy to calculate an upper bound on the number of nodes reached, when duplicate nodes are ignored

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Forwarding in a Finite Network

n11 n12 n13 n14

n1 n2 n3 n4

n21 n22 n23 n24

n0

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The pdf Algorithm

Input of the algorithm:–n: number of nodes in the network–c: number of nodes to which a node forwards a

message– f: probability with which a node forwards a message–l: number of levels of message forwarding

Output of the algorithm:–pdf for the number of nodes reached at level l–expected number of nodes reached at level l–pdf for the number of nodes reached up through level l–expected number of nodes reached up through level l

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Probability of a Duplicate Node

Let N be a set of nodes having cardinality n A be a subset of N having cardinality a B be a subset of N having cardinality b where a ≤ b

The pdf p(k), 0 ≤ k ≤ a, that A ∩ B has cardinality k, is given by:

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Calculating the pdfs

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Dl-1Sl-1

Cl k

SlDl

sprev[j] scurr[j]

pdf of number of distinct nodesat level l-1 and all prior levels

pdf of number of new nodes at level l -1

pdf of numberof new nodes

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Upper Bound Analysis

Upper bound on the number of nodes reached at level lUBs = min(cl, n)

Upper bound on the number of nodes reached up through level lUBd = min(1 + c + c2 + … + cl , n)

= min( (c l+1 – 1) / (c – 1), n) if c > 1

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Nodes Reached Upper Bound vs. pdf Analysis

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c=4

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pdfs for Total Nodes Reached Varying c

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l = 10

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pdfs for Total Nodes Reached Varying l

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c=4

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pdfs for Total Nodes Reached Varying f

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c = 4l = 10

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Related WorkS. E. Deering and D. R. Cheriton, “Multicast routing in

datagram internetworks and extended LANs,” ACM Trans. Computer Systems, vol. 8, no. 2, pp. 85-110, 1990

Z. J. Hass, J. Y. Halpern and L. Li, “Gossip-based ad hoc routing,” IEEE/ACM Trans. Networking, vol. 14, no. 3, pp. 479-491, June 2006

S. M. Hedetniemi, S. T. Hedetniemi and A. L. Liestman, “A survey of gossiping and broadcasting in communications networks,” Networks, vol. 18, pp. 319-349, 1988

D. Shah, “Gossip algorithms,” Foundations and Trends in Networking, vol. 3, no. 1, pp. 1-125, 2008

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ConclusionsThe expected numbers of nodes reached up through

a given level, and at a given level, are substantially less than those for the upper bound analysis

The pdfs for the number of nodes reached up through a given level, and at a given level, exhibit a wide range, particularly for smaller values of the–Degree of forwarding–Probability of forwarding

As the forwarding probability decreases, the expected number of nodes reached decreases quite rapidly

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Future Work

Neighborhoods– fully connected within neighborhoods but

only partially connected between neighborhoodsFaulty nodes and linksTime to reach a certain number of nodesEnergy and power consumption at nodesApplication to existing iTrust system

–decentralized publication, search and retrieval

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Questions? Comments?

Louise Moser - moser@ece.ucsb.eduMichael Melliar-Smith - pmms@ece.ucsb.edu

This work was supported in part byNational Science Foundation Grant

NSF CNS 10-16193

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